User controls are basically designed so that a web developer can create a composite control to be reused in an application.
Unfortunately, there is no immediately obvious replacement for the classic asp method of an include file that declares a "global" variable such as a db connection so that any page may include it.
A more advanced technique is now available with .Net because of the much better language support in ASP.Net applications. The basic idea is simply to construct a base class that inherits from System.Web.UI.Page. Then you create a Protected property on that class that provide access to your DB connection. In that class you can set up the connection in the Page.Load event handler and close the connection in the Page.UnLoad handler. Then all of your asp.net application's ASPX pages inherit from the base class you created instead of System.Web.UI.Page. This provides them with the standard web form class functionality but also the additions you made.
I have written up a slightly more comprehensive discussion on this idea. You can find it
here.
Peter
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Work smarter, not harder.